ATON is a hopefully evolving classification theory. It aspires to unify knowledge around numbers and prefers naive methods. Some of the older posts are wrong but I'll keep them for the sake of continuity.

Saturday, November 11, 2006

Multifactorials

Lately I found, to my great surprise, that ATON Exponential Set is in fact already defined in the number domain as the second multifactorial, also called Barnes G-function;

G2(z+1)=Gamma(z)*G2(z)
G2(1)=1
Gamma(z+1)=z*Gamma(z)

Higher multifactorials follow. To see how it works, expand 3 to depth 6;

G0(3) = 3 = 3*2^0
G1(3)=3.2.1.1 = 6 = 3*2^1
G2(3)=(3.2.1.1).(2.1.1).(1.1).(1) = 12 = 3*2^2
G3(3)=[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)] = 24 = 3*2^3

G4(3)={[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]} = 48 = 3*2^4

G5(3)={{[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]}}.

.{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}} = 96 = 3*2^5

G6(3)={{{[(3.2.1.1).(2.1.1).(1.1).(1)].[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.
.{[(1.1).(1)].[(1)]}.
.{[(1)]}}.
.{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(2.1.1).(1.1).(1)].[(1.1).(1)].[(1)]}.{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(1.1).(1)].[(1)]}.{[(1)]}}.
.{{[(1)]}}}.

.{{{[(1)]}}} = 192 = 3*2^6

Therefore Gn(3)=3*2^n.

Instead of multiplying, one could add the expanded elements (with no apparent benefits);

GS0(3)=3
GS1(3)=7
GS2(3)=14
GS3(3)=25
GS4(3)=41
GS5(3)=63
GS6(3)=92

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